One-sided Maximal Functions Research Articles

Two weighted norm inequalities for generalized one-sided maximal function on one-sided weighted like Morrey space

Two weighted norm inequalities for generalized one-sided maximal function on one-sided weighted like Morrey space

Characterizations of parabolic Muckenhoupt classes

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of p-Laplace type. The main results include characterizations for the limiting parabolic A∞ and A1 classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón–Zygmund decompositions, covering and chaining arguments appear in the proofs.

A Unified Version of Weighted Weak Type Inequalities for One-Sided Maximal Function on $${\mathbb {R}}^2$$

A Unified Version of Weighted Weak Type Inequalities for One-Sided Maximal Function on $${\mathbb {R}}^2$$

Some quantitative one-sided weighted estimates

In this paper we provide some quantitative one-sided estimates that recover the dependences in the classical setting. Among them we provide estimates for the one-sided maximal function in Lorentz spaces and we show that the conjugation method for commutators works as well in this setting.

Weighted inequalities for higher dimensional one-sided Hardy–Littlewood maximal function in Orlicz spaces

In this paper, weighted extra-weak and weak type inequalities have been characterized for the one-sided Hardy–Littlewood maximal function on the plane. We have addressed conditions on pair of weights for which the dyadic one-sided maximal function on higher dimension is locally integrable. In the process, we characterize weights for which the one-sided Hardy-Littlewood maximal function satisfies restricted weak type inequalities on the plane, thus extending the result of Kerman and Torchinsky to the one-sided Hardy-Littlewood maximal function.

Regularity of Commutators of the One-Sided Hardy-Littlewood Maximal Functions

In this paper, the regularity properties of two classes of commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants are investigated. Some new bounds for the derivatives of the above commutators and the boundedness and continuity for the above commutators on the Sobolev spaces will be presented. The corresponding results for the discrete analogues are also considered.

On the regularity of one-sided fractional maximal functions

Abstract In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators ℳ β + $ \mathcal{M}_{\beta}^{+} $ and ℳ β - $ \mathcal{M}_{\beta}^{-} $ map W 1 , p ⁢ ( ℝ ) $ W^{1,p}(\mathbb{R}) $ into W 1 , q ⁢ ( ℝ ) $ W^{1,q}(\mathbb{R}) $ with 1 <p <∞, 0≤β<1/p and q=p/(1-pβ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators M β + $ M_{\beta}^{+} $ and M β - $ M_{\beta}^{-} $ from ℓ 1 ⁢ ( ℤ ) $ \ell^{1}(\mathbb{Z}) $ to BV ⁢ ( ℤ ) $ {\rm BV}(\mathbb{Z}) $ . Here BV ⁢ ( ℤ ) $ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces Ws,p(R2) for 0≤s≤1 and 1<p<∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from l1(Z2) to BV(Z2). Here BV(Z2) denotes the set of all functions of bounded variation on Z2.

Weighted and vector-valued inequalities for one-sided maximal functions

In this paper, we study weighted and vector-valued inequalities for one-sided maximal functions. In particular, we give a new proof of lr-valued extension of weighted Lp-inequalities for one-sided maximal functions. In the process, we prove an analogue of the well-known Fefferman–Stein’s weighted lemma in the context of one-sided maximal functions. Further, we also study lr-valued extension of the lemma.

Sharp maximal estimates for BMO martingales

We introduce a method which can be used to study maximal inequalities for martingales of bounded mean oscillation. As an application, we establish sharp $\Phi$-inequalities and tail inequalities for the one-sided maximal function of a BMO martingale. The results can be regarded as BMO counterparts of the classical maximal estimates of Doob.

A weak-type inequality for the martingale square function

Let f=(fn)n≥0 be a real-valued martingale satisfying f0≥0 almost surely and let S(f) denote the square function of f. The paper contains the proof of the weak-type bound λP(S(f)≥λ)≤eEsupn≥0fn,λ>0, involving the one-sided maximal function on the right-hand side. The constant e is the best possible.

MIXED NORM INEQUALITIES FOR SOME DIRECTIONAL MAXIMAL OPERATORS

AbstractMixed norm inequalities for directional operators are closely related to the boundedness problems of several important operators in harmonic analysis. In this paper we prove weighted inequalities for some one-dimensional one-sided maximal functions. Then by applying these results, we establish mixed norm inequalities for directional maximal operators which are defined from these one-dimensional maximal functions. We also estimate the constants in these inequalities.

A boundedness criterion for general maximal operators

We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a reversed weak type inequality, we obtain a boundedness criterion for MB on an arbitrary quasiBanach function space X. Being applied to specific B and X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M + on L p.

Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

Let $f=(f_n)_{n\geq 0}$ be a nonnegative submartingale starting from $x$ and let $g=(g_n)_{n\geq 0}$ be a sequence starting from $y$ and satisfying $$|dg_n|\leq |df_n|,\quad |\mathbb{E}(dg_n|\mathcal{F}_{n-1})|\leq \mathbb{E}(df_n|\mathcal{F}_{n-1})$$ for $n\geq 1$. We determine the best universal constant $U(x,y)$ such that $$\mathbb{P}(\sup_ng_n\geq 0)\leq ||f||_1+U(x,y).$$ As an application, we deduce a sharp weak type $(1,1)$ inequality for the one-sided maximal function of $g$ and determine, for any $t\in [0,1]$ and $\beta\in\mathbb{R}$, the number $$ L(x,y,t,\beta)=\inf\{||f||_1: \mathbb{P}(\sup_ng_n\geq \beta)\geq t\}.$$ The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.

One‐sided operators inLp(x)spaces

AbstractBoundedness of one‐sided maximal functions, singular integrals and potentials is established inL(I) spaces, whereIis an interval inR. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Maximal Function Estimates for the Parabolic Mean Value Kernel

We obtain parabolic and one-sided maximal function estimates for nonisotropic dilations of the mean value kernel for the heat equation ......

On the uniqueness of the one-sided maximal functions of Borel measures

On the uniqueness of the one-sided maximal functions of Borel measures

Weak weighted inequalities for a dyadic one-sided maximal function in $ \mathbb {R}^{n}$

In this note we introduce a dyadic one-sided maximal function defined as formula math. where Q + is a certain cube associated with the dyadic cube Q and f ∈ L 1 loc (R n ). We characterize the pair of weights (w,v) for which the maximal operator M +,d applies L p (v) into weak-L p (w) for 1 ≤ p < oo.

Norm inequalities relating one-sided singular integrals and the one-sided maximal function

AbstractIn this paper we prove that if a weight w satisfies the condition, then the Lp(w) norm of a one-sided singular integral is bounded by the Lp(w) norm of the one-sided Hardy-Littlewood maximal function, for 1 &lt; p &lt; q &lt; ∞.

Weighted inequalities for one-sided maximal functions in Orlicz spaces

Weighted inequalities for one-sided maximal functions in Orlicz spaces